Analytical Modeling and Analysis of Workload Allocation in a Network of Service Centers

Transcription

1 Jurnal Teknolog Maklumat & Multmeda 3(006): 17-9 Analytcal Modelng and Analyss of Workload Allocaton n a Network of Servce Centers RAHELA RAHIM & KU RUHANA KU MAHAMUD ABSTRACT Redesgn of busness processes s currently attractng a lot of nterest. Although many tools and technque have been used to model such processes, less attenton s gven to the analytcal aspect to support or optmze the redesgn process. In other applcaton areas, where a network of server systems are concerned such as computer and communcaton systems, telecommuncaton systems and manufacturng systems, many analytcal technques have been developed to analyze the temporal aspect of the systems. The smlartes exst between these knds of system and busness processes. An analytc approach s presented that consders the performance of busness process desgns. A close-loop expresson s derved from a non-lnear optmzaton problem to obtan an optmal value for jobs arrval. Keywords: Analytcal modelng; optmzaton and queueng system; workload allocaton ABSTRAK Ketka n reka bentuk semula bag proses kerja menark mnat banyak phak. Walaupun banyak alatan dan teknk dgunakan untuk memodel proses, kurang perhatan dberkan pada aspek analtkal untuk menyokong atau mengoptmumkan reka bentuk semula proses. Dalam bdang aplkas selan darpada bdang d mana suatu rangkaan sstem-sstem pelayan sepert sstem pengkomputeran dan komunkas, sstem telekomunkas dan sstem perndustran, banyak teknk analtkal telah dbangunkan untuk menganalss aspek masa bag sstem. Terdapat kesamaan d antara sstem-sstem n dengan proses kerja. Pendekatan analtk dcadangkan bag mempertmbangkan prestas proses kerja pada perngkat reka bentuk proses. Suatu ungkapan gelung tertutup dtahkkkan darpada masalah pengoptmuman bukan-lnear untuk mendapatkan nla optmum bag ketbaan kerja. Kata kunc: Permodelan beranaltkal; pengoptmuman dan sstem bars; penuntukan beban kerja 0-Rahela 17 4/6/05, 9:01 AM

2 INTRODUCTION Recently busness process redesgns have been gven a lot of attenton. Snce then many modelng technques and tools have been used to support the redesgn process. However, most of the currently avalable tools usng statc model such as dagram to model process. Some are qute dynamc where the functonal aspect of the process have been modeled usng smulaton. However hardly any studes are found n the lterature that use analytcal model to study the quanttatve behavor and optmzaton of busness process. Generally for busness processes, the cost of the process depends on ts tme behavor. Customer satsfactons also often depend on tme-based performance measure such as response tme or turnaround tme. For these reasons, t s mportant to optmze the quanttatve performance of busness process. An mportant advantage of analytcal modelng s that they allow for the off-lne analyss of the effect of redesgn process, wthout dsturbng actual processes. To obtan the effectve desgn model, feedback to the desgner concernng process functonal and alternatve desgn opton should be done early n the desgn process. At ths stage, analytcal modelng provdes quanttatve propertes, whereby these wll provde the global ndcaton of the expected performance. In the fnal stage, more accurate predctons may be requred to fne-tune desgn. Therefore, the analytcal modelng proposed here s at the hghest abstracton level of desgn process,.e. to get the frst dea on how the process should behave. The quanttatve measures of busness processes to those of concurrent dscrete-event system have been focused. Based on the quanttatve measure of busness proceses to those of concurrent dscrete-event system, shows that by usng quanttatve modelng, arrval to resource center can be reallocated to get the optmal performance measure. The ssue of job allocaton n a network of servce centers where dfferent servce centers have dfferent job processng tme have been focused. The optmzaton crteron studed here s to mnmze the expected job response tme n the systems to whch jobs are allocated. Jobs arrve at a scheduler that allocate jobs to the servce center accordng to a calculated arrval rate computed usng Lagrange multpler theorem. The paper s organzed as follows: next secton shows the related work n ths area followed by the proposed queueng and optmzaton model. Later numercal results and models verfcaton are presented. Fnally concludng remarks and drectons for further research are descrbed. RELATED WORK The problem of workload allocaton s common to a varety of busness systems especally when t nvolves a network of servce centers. Workload allocaton seeks to allocate job arrval among servce centers as evenly as 18 0-Rahela 18 4/6/05, 9:01 AM

3 possble. In a parallel settng, and partcularly for mult servce center systems where jobs may have many possble paths at the job s scheduler, jobs allocaton problem s of nterest, each job enterng the scheduler, a path s assgned to optmze the allocaton of workload (Ln & Kumar 1984; N & Hwang 1985; Lu 1999). The workload allocaton problem s of partcular nterest for networks of servce centers, snce there are several ways that affect the dstrbuton of workload among the centers (Ross & Yao 1991). Queueng network models have been recognzed as powerful tools for evaluatng performance of computer systems (Gelenbe & Mtran 1980; Allen 1990) and communcaton network (Klenrock 1975; Harrson & Patel 199; Bell & Wllams 1999; Gunther 000; Menasce & Almeda 000). Ths analytcal model has become a very mportant tool for predctng the behavour of new desgns or proposed changes to exstng systems (Kouvatsos & Othman 1986, 1989; Hsao & Lazar, 1990, 1991; Smth & Wllams 001). Most queueng network models are used ether by makng assumptons to assure exact numercal soluton or by employng approxmate method (Kobayash 1974; Chandy et al. 1975; Harrson & Patel 199). Lazar (1981, 1983, 1984), consdered the control of arrvals to a network of queues wth the objectve of maxmzng throughput subject to a response tme constrant. He developed a throughput tme delay functon based on an optmalty crteron where the arrval that maxmze the throughput under the constrant of the average response tme wll not exceed a preassgned value. Then contnued wth the problem of random routng (Kouvatsos & Othman 1989). All these lterature have been devoted to the probablstc analyss of queueng system, ther optmzaton s somewhat laggng behnd. Network of queues n parallel are a natural way to model problems of resource and traffc allocaton and many optmzaton problems based on them have been studed for nstance n Klenrock (1975); N & Hwang (1985); Boxma (1995); Chombe & Boxma (1995); Koole (1999); and de Jongh (00). Most of the studes focus on reducng the amount of watng tme n a system wth several servers ether parallel or seral. However, none of the studes consdered the mpact of jobs nter arrval and servce tme varaton n modelng the systems performance. Although analytcal workload allocaton models are wdely appled n area of computer and communcaton network, they are also applcable n the case of busness processes especally by the ncreasng applcaton of workflow management systems. QUEUEING SYSTEM MODEL When several users request for a job (e.g customer request for product or servce) at the same tme, competton for the use of a common resource and the lmted capacty of the resource can gve rse to congeston, hence queueng s a common phenomena. Queueng occurs normally when the demand exceeds the servce capacty of the resource and even when the 19 0-Rahela 19 4/6/05, 9:01 AM

4 otherwse occur. Ths s due to the fact that the nter arrval tmes of the request, and ther requred servce tmes, are generally not fxed, therefore a mathematcal model of congeston represents nter arrval and servce tmes of users by random varables. Queueng Theory s devoted to the descrpton, analyss and optmzaton of such queueng system (Boxma 1995). It focuses on a few key performance measures, lke queue lengths and watng tmes. Due to the stochastc nature of the arrval and servce processes, and of the allocaton process of jobs through a network of queues, the man performance measures are also random varables. Wth ths n mnd, we use multple queue multple server model to represent a central job allocaton system as shown n Fgure 1. In usng ths model, resources that provde servces are represented by FIGURE 1. Multple queue multple server model servce centers at whch jobs queue and compete for servce. The workload s modeled as a sngle stream of job. A common example of workload s order request. Total workload rate arrved s gven by ϕ. Each newly arrved job, t s assgned to a node or server wth rate λ, accordng to the schedulng polcy used. We consder the set of servce centers to be heterogeneous whch s common cases n real systems and also t can be generalzed to homogeneous servers. In the contex of general queueng network models, the generalzed exponental (GE) dstrbutonal model of the form (Kouvatsos & Othman 1986, 1989). C t fs ( t) = 1 4μ μ u ()+ t exp,t 0 0 C + 1 C + 1 C + 1 ( ) Where μ s the mean servce rate, C s the coeffcent of varaton and u 0 (t) s the unt mpulse functon, has been used to represent the nter arrval and servce tme dstrbuton. Ths model s robust and versatle due to t memoryless propertes and has been shown to maxmze the entrophy 0 0-Rahela 0 4/6/05, 9:0 AM

5 functon subject to mean value constrants. Furthermore t has been shown n (Kouvatsos 1986; Ku Mahamud 1993) that the exact mean number of jobs n the GE/GE/1 queue s gven by ρ Ca + ρc L = ρ s, for 1 C 1 C a s ρ < 1 Where Ca, C are the squared coeffcents of varaton of the nter s arrval and servce tme dstrbutons respectvely. Ths mean number of jobs functon wll be used as the objectve functon n the optmzaton method. OPTIMIZATION MODEL USING GENERALIZED EXPONENTIAL (GE) DISTRIBUTION In ths secton, the use of Lagrange method to GE type dstrbutonal model s proposed. In ths case an optmzaton problem of queueng system can be generalzed to a number of arrval and servce dstrbuton by confgurng the value of coeffcent of varaton for nter arrval and servce tme. We formulated an optmzaton problem of N -GE/GE/1 where N ndcates a number of queueng system, as below: Notaton: λ : mean arrval rate of job- at servce center μ : mean servce rate of servce center β : mean servce tme of servce center φ : total arrval rate L : mean jobs queue length C s : mean coeffcent of varaton n servce tme C a : mean coeffcent of varaton n nter arrval tme ρ : utlzaton of servce center P1 Mn N L = N = 1 = 1 Cs λβ 1 Ca + 1 D + λβ 1 λβ (1) N s.t λ = φ = 1 () 1 0 λ, = 1, K, N. (3) β 1 0-Rahela 1 4/6/05, 9:0 AM

6 λ 0 (4) β 0 (5) λ where ρ = and β μ 1 = μ Problem P1 allows an analytcal soluton. Usng Lagrange multpler theorem we obtan wth δ the Lagrange multpler, the followng frst order Kuhn-Tucker constrants: d λβ Cs 1 Ca + 1 D λβ + δ dλ 1 λβ = = 1,,N (6) N λ = φ = 0 (7) = 1 From (6) and (7) we fnd the unque optmal values 1 Cs + C a λ* = 1 β Cs 1+ μδ 1 (8) and Lagrange multpler s derved by solvng the constrant equaton below: N = 1 1 C 1 s + C a β Cs 1+ μδ 1 φ = (9) When C a =1 and, C s =1, the GE traffc optmal expresson s reduced to exponental optmal expresson. D s the cost assocated wth havng one job n queue and for smplcty, we assgn the value of 1. COMPUTATIONAL RESULT In ths secton, numercal results are presented to assess the credblty of the GE dstrbuton used. Two confguratons wll be shown. For the frst confguraton, servce rate of the tasks are assumed to be μ1 = 3, μ = 4, Ca = 05., C = 03., = 0., = a C C s1 s. The mprovement of the performance measures s presented n Fg. and 4. To verfy the results, we use smulaton and the comparatve results are presented n Fgure 3 and 5. 0-Rahela 4/6/05, 9:03 AM

7 Table 1 provdes the result of mean queue length and mean response tme obtaned from the classcal and proposed allocaton scheme of two GE/ GE/1 queueng systems wth servce rate μ 1 = 3, μ = 4, and varaton parameters of nter arrval and servce tme, Ca 1 = 0.5, Ca = 0.3 and Cs 1 = 0., Cs = 0.4. TABLE 1. Result of classcal and proposed approaches of -GE/GE/1 queueng system, L: Mean Queue Length, W: Mean Response Tme Classcal Proposed Classcal Proposed λ 1 λ λ 1 λ L W L W FIGURE. Performance mprovement of mean queue length for a dual GE/GE/1 queueng system Fgure and 4 shows the comparatve mprovement n the mean number of jobs and mean response tme n the -GE/GE/1 systems obtaned n Table 1. The maxmum mprovement for the gven dataset s gven by 13.5%. The results obtaned n Table 1 are valdated usng smulaton model whereby the same parameters settng have been used. The maxmum comparatve error of 1.08% s gven n Fgure 3 and 5. Other parameter settng of two GE/GE/1 queueng systems wth servce rate μ 1 = 3, μ = 4, and varaton parameters of nter arrval and servce tme, Ca 1 = 0.1, Ca = 0. and Cs 1 = 0.4, Cs = 0.3 s presented n Table. Fgure 6 and 7 shows the comparatve mprovement n the mean queue length and mean response tme of the -GE/GE/1 systems obtaned n Table. The maxmum mprovement for the gven dataset s gven by 7.14%. 3 0-Rahela 3 4/6/05, 9:03 AM

8 FIGURE 3. Analytcal versus smulaton result FIGURE 4. Performance mprovement of mean response tme for a dual GE/GE/1 queueng system FIGURE 5. Analytcal versus smulaton result 4 0-Rahela 4 4/6/05, 9:03 AM

9 TABLE. Result of classcal and proposed approaches of GE/GE/1 queueng system Classcal Proposed Classcal Proposed λ 1 λ λ 1 λ L W L W The results obtaned n Table are valdated usng smulaton model whereby the same parameters settng have been used. The maxmum comparatve error of 13.06% s gven n Fgure 7 and 9. FIGURE 6. Performance mprovement of mean queue length for a dual GE/GE/1 queueng system FIGURE 7. Analytcal versus smulaton result 5 0-Rahela 5 4/6/05, 9:03 AM

10 Further analyss for sample cases of number of servers N = {3, 4, 5, 6}, are shown below. FIGURE 8. Performance mprovement of mean response tme for a dual GE/GE/1 queueng system FIGURE 9. Analytcal versus smulaton result Numercal Result for Three Queueng Systems: μ = ( 31,, ) C = ( a ,.,. ) C = ( ) s ,.,. Numercal Result for Four Queueng Systems μ = ( 4, 31,, ) C = ( ) a ,.,.,. C = ( ) s 004.,., 0301.,. Numercal Result for Fve Queueng Systems μ 5431,,,, C a ,.,.,.,. C s ,.,.,.,. Numercal Result for Sx Queueng Systems μ = ( 654,,, 31,, ) C a = ( ,.,..,., 0306.,. ) C ,.,., ,.,. s = ( ) = ( ) = ( ) = ( ) The analyss shows that a larger range for the servce rates results n greater percentage mprovements of our aggregate objectves. The result of 6 0-Rahela 6 4/6/05, 9:04 AM

11 the analyss for the queueng systems s summarzed n Fgure 10 below. Utlzaton of 90% has been assumed as ths ndcates that the servce centers are almost fully utlzed. FIGURE 10. Performance mprovement for a sample number of servers where ρ = 0.9. CONCLUSION A new optmzaton model of allocatng job arrvals to a network of servce centers based on Generalzed Exponental arrval and servce tme dstrbuton has been proposed. A closed-loop expresson to obtan the routng rate was descrbed. Analytcal model and smulaton approaches were used to show that the classcal allocaton of total arrvals between two servce centers wth the same utlzaton rate does not provde an effcent performance result for the queueng systems. The result for a number of servce centers was shown to portray the mprovement. The GE dstrbuton has been employed because t could represent exponental and other general dstrbuton. The analytcal modelng proposed can be used to gude the busness process redesgn n some parts where networks of servce centers are concerned. There are several drectons to extend the applcablty of ths allocaton algorthm such as dfferent performance objectve functon, other arrval and servce dstrbuton and arrval wth dfferent type of jobs. These examples would nvolve nterestng mathematcal problem and could be the subject of future research. REFERENCES Allen, A. O Probablty, statstcs and queueng theory wth computer scence applcatons. nd ed. San Dego: Academc Press. Bell, S. L. & Wllams, R. J Dynamc schedulng of a system wth two parallel servers: Asymptotc optmalty of a contnuous revew threshold polcy n heavy Trac. In Proceedngs of the 38th Conference on Decson and Control, Pheonx, Arzona: Rahela 7 4/6/05, 9:04 AM

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13 Lu, J. B A multlevel load balancng algorthm n a dstrbuted system. Proceedngs of the 19th annual conference on Computer Scence: Menasce, D. A. & Almeda, V. A. F Scalng for E-Busness. Prentce Hall. N, L. M. & Hwang, K Optmal load balancng n a multple processor system wth many job classes. IEEE Trans. Software Engneerng: Ross, K. W. & Yao, D. D Optmal load balancng and schedulng n a dstrbuted computer system. Journal of the ACM 38(3): Smth, C. U. & Wllams, L. G Performance solutons, a practcal gude to creatng responsve, scalable software. Pearson educaton. Tantaw, A. N. & Towsley, D Optmal statc load balancng n dstrbuted computer systems. J. ACM 3(): Rahela Rahm Fakult Sans Kuanttatf UUM Sntok Kedah rahela@uum.edu.my Ku Ruhana Ku Mahamud Pengarah Pusat Pengajan Sswazah UUM Sntok Kedah ruhana@uum.edu.my 9 0-Rahela 9 4/6/05, 9:04 AM